On Sat, 20100130 at 12:28 +1300, Rob Freeman wrote:
> Chris,
>
> On Fri, Jan 29, 2010 at 3:16 PM, Christopher Menzel <cmenzel@xxxxxxxx>
> wrote:
> >
> > ... On reflection, I will qualify my original claim that Russell's
> > paradox has *nothing* to do with undecidability. For there is in
> > fact a similarity between the usual argument given in presentations
> > of Russell's paradox and the method of diagonalization in
> > computability theory  a method often used in proofs of
> > undecidability. Russell's argument can be used more generally to
> > show that, for any given set A, the set of all nonselfmembered
> > members of A cannot itself be a member of A and the proof is a clear
> > instance of diagonalization. The paradox itself ensues from this
> > theorem if one also assumes (or one is able to prove) that there is
> > a universal set.
> >
> > So there is a similarity between the method of proof in Russell's
> > paradox and a common method of proof in computability theory. But,
> > conceptually, the paradox itself has nothing to do with
> > undecidability/incompleteness/computability per se.
>
> I'm not sure what you mean by
> "undecidability/incompleteness/computability per se." (01)
It means the paradox has nothing to do with the concepts themselves.
Russell's paradox is a paradox of set theory; it was discovered decades
before computability theory was even formulated and none of the concepts
of computability theory are presupposed by the paradox in any way. Nor
does it have any implications for them. The only connection, as I
pointed out, is that the *structure* of the usual proof of Russell's
paradox (more accurately, the usual proof of a proposition that, with
the assumption of a universal set, entails Russell's paradox) is similar
to the structure of some proofs of undecidability. (02)
> I'm happy with your agreement that "there is in fact a similarity
> between the usual argument given in presentations of Russell's paradox
> and the method of diagonalization in computability theory  a method
> often used in proofs of undecidability." (03)
My agreement?? Again you are being entirely disingenuous. For X to
agree with Y about A implies that Y has asserted A; and it typically
presupposes that Y also understands A. So to characterize my
observation about diagonalization in the proof of Russell's paradox as
something I have *agreed* with you about suggests that it is something
that you yourself have asserted and that you understand. You've
certainly not asserted it, and the numerous confusions and errors about
logic and computability in your own posts lead one, at the least, to be
suspicious of whether you understand it. (04)
cm (05)
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